A new pinching theorem for complete self-shrinkers and its generalization

• 作者: Hongwei Xu Li Lei Zhiyuan Xu
• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2020,（6）
• 页码范围: 1139-1152
• 页数: 14页
• 语种: 英文

摘要

In this paper,we firstly verify that if Mn is an n-dimensional complete self-shrinker with polynomial volume growth in Rn+1,and if the squared norm of the second fundamental form of M satisfies 0 ≤ S-1 ≤ 1/18,then S ≡ 1 and M is a round sphere or a cylinder.More generally,let M be a complete γ-hypersurface of codimension one with polynomial volume growth in Rn+1 with λ ≠ 0.Then we prove that there exists a positive constant γ such that if |γ| ≤ γ and the squared norm of the second fundamental form of M satisfies 0 ≤ S-3λ ≤ 1/18,then S ≡ βλ,λ ＞ 0 and M is a cylinder.Here βλ =1/2(2 + λ2 + |λ|√γ2+4).